Term for poetic technique in which the last word of a line is the first word of the next

"Runs in the family" by Amanda Palmer contains the following lyrics:

Strips in the city and shares all her best tricks with

Me?

Well, I'm well

The first word of the bridge, "Me?", is also the last word in the preceding verse. However, unlike with anadiplosis where the word used in both sentences is repeated, here the word is only used once.

This may be a form of enjambment, but it seems slightly different to the examples I've come across. In particular, the three lines are not strictly well formed in conjunction, as the "me" is used both as the start and end of a sentence depending on which pair of sentences it is considered to belong. This is contrary to many examples of enjambment in which each line follows the previous semantically.

Is this difference a meaningful distinction, and if so, does the technique have a name?

To my knowledge, the most precise word for the device she is using is simply enjambment. Enjambment itself isn't concerned with how a particular word is used in a given line, only how the line structure is broken up.

From Wikipedia:

In poetry, enjambment is incomplete syntax at the end of a line; the meaning runs over from one poetic line to the next, without terminal punctuation.

That said, she is using enjambment to play with the ambiguity of which sentence the word belongs in. I don't think this is contrary to enjambment, but just one of the effects that it helps create in a poem. Perhaps more common uses are to achieve a certain meter, emphasize/de-emphasize rhymes or subvert expectations, but once again, those are just effects of enjambment, not the definition.